Properties

Label 9450.3449
Modulus $9450$
Conductor $945$
Order $18$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,9,15]))
 
pari: [g,chi] = znchar(Mod(3449,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(945\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{945}(614,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.dl

\(\chi_{9450}(299,\cdot)\) \(\chi_{9450}(1949,\cdot)\) \(\chi_{9450}(3449,\cdot)\) \(\chi_{9450}(5099,\cdot)\) \(\chi_{9450}(6599,\cdot)\) \(\chi_{9450}(8249,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((9101,6427,6751)\) → \((e\left(\frac{7}{18}\right),-1,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(3449, a) \) \(1\)\(1\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(-1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(3449,a) \;\) at \(\;a = \) e.g. 2